## On operators with an absolute value condition.(English)Zbl 1076.47015

Let $$A$$ be a bounded linear operator on a complex Hilbert space $$H$$. An operator $$A$$ is said to be a class A operator if and only if $$| A| ^{2}\leq | A^{2}|$$ holds. An operator $$A$$ is said to be a polynomially class A operator if $$p(A)$$ is in class A with a nontrivial polynomial $$p$$. It is well-known that class A contains the class of $$p$$-hyponormal operators for $$p>0$$.
The authors show that (i) class A operators are finitely ascensive, (ii) $$A \otimes B$$ belongs to class A if and only if $$A$$ and $$B$$ belong to class A, and (iii) Weyl’s theorem holds for $$f(A)$$ if $$A$$ is a polynomially class A operator and $$f$$ is an analytic $$H$$-valued function on a neighborhood of $$\sigma(A)$$.
Result (ii) is an extension of the second author’s [Glasg. Math. J. 42, 371–381 (2000; Zbl 0990.47017)] in which the case that $$A$$ is $$p$$-hyponormal is treated. Result (iii) completely extends [B. P. Duggal and S. V. Djordjević, Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 40, 49–56 (2000; Zbl 0997.47019)] and [Y. M. Hand and W. Y. Lee, Proc. Am. Math. Soc. 128, 2291–2296 (2000; Zbl 0953.47018)] through slightly different approaches.

### MSC:

 47B20 Subnormal operators, hyponormal operators, etc. 47A80 Tensor products of linear operators

### Citations:

Zbl 0990.47017; Zbl 0997.47019; Zbl 0953.47018
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